Question

Find h'(x) where f(x) is an unspecified differentiable function. h(x) = f(x)/x⁶​

Answer 1

To find h'(x) for h(x) = f(x)/x⁶, we use the quotient rule for differentiation. The result after simplification is h'(x) = (f'(x) - 6f(x)/x)/x⁶.

Step-by-step explanation:

To find h'(x), which is the derivative of h(x) with respect to x, when h(x) = f(x)/x⁶, we will use the quotient rule for differentiation. The quotient rule states that if we have a function of the form u(x)/v(x), the derivative of this function with respect to x is given by:

(u'(x)v(x) - u(x)v'(x))/(v(x))^2

Therefore, in this case, u(x) = f(x) and v(x) = x⁶. The derivative of f(x) is f'(x) (this is given because f(x) is specified as differentiable), and the derivative of x⁶ is 6x⁵.

Applying the quotient rule:

h'(x) = (f'(x)x⁶ - f(x)6x⁵)/(x⁶)^2

Simplifying the expression, we get:

h'(x) = (f'(x)x⁶ - 6f(x)x⁵)/(x¹²)

h'(x) = (f'(x) - 6f(x)/x)/x⁶

This is the derivative of h(x) expressed as h'(x).

Answer 2

a. The derivative of the function h(x) = f(x)/x⁶ with respect to x is h'(x) = -6f(x)/x⁷ + f'(x)/x⁶.

Step-by-step explanation:

a. To find the derivative h'(x) of the given function h(x) = f(x)/x⁶, we use the quotient rule. The quotient rule states that if h(x) = g(x)/k(x), then h'(x) = (g'(x)k(x) - g(x)k'(x))/(k(x))². In this case, g(x) = f(x) and k(x) = x⁶. Applying the quotient rule, we get h'(x) = (f'(x)x⁶ - 6f(x)x⁵)/(x⁶)² = -6f(x)/x⁷ + f'(x)/x⁶.

Understanding the quotient rule is essential in calculus, as it provides a method for finding the derivative of a quotient of two functions. The rule helps handle situations where a function is expressed as the ratio of two other functions. In this case, the derivative h'(x) involves the derivative of the numerator f(x) and the derivative of the denominator x⁶, applying the power rule and the constant multiple rule as necessary.

In summary, the derivative h'(x) of the function h(x) = f(x)/x⁶ is calculated using the quotient rule, resulting in h'(x) = -6f(x)/x⁷ + f'(x)/x⁶. This expression provides the rate of change of h(x) with respect to x and is a fundamental concept in calculus.